Sufficient conditions for smoothing codimension one foliations

Author:
Christopher Ennis

Journal:
Trans. Amer. Math. Soc. **276** (1983), 311-322

MSC:
Primary 57R30; Secondary 57R10, 58F18

DOI:
https://doi.org/10.1090/S0002-9947-1983-0684511-4

MathSciNet review:
684511

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact manifold. Let be a nonsingular vector field on , having unique integral curves through . For continuous, call whenever defined. Similarly, call .

For , a foliation of is said to be smoothable if there exist a foliation , which approximates , and a homeomorphism such that takes leaves of onto leaves of .

Definition. A transversely oriented Lyapunov foliation is a pair consisting of a codimension one foliation of and a nonsingular, uniquely integrable vector field on , such that there is a covering of by neighborhoods , , on which is described as level sets of continuous functions for which is continuous and strictly positive.

We prove the following theorems.

Theorem 1. *Every* *transversely oriented Lyapunov foliation* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

Theorem 2. *If* *is a* *transversely oriented Lyapunov foliation, with* *and* *continuous for* *and* , *then* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the version of Theorem 1.

Theorem 3. *If* *is a* *transversely oriented Lyapunov foliation, with* *and* *is continuous, then* *is* *smoothable to a* *transversely oriented Lyapunov foliation* .

**[1]**A. Denjoy,*Sur les courbes définies par les équations differentielles à la surface du tore*, J. Math. Pures Appl.**11**(1932), 333-375.**[2]**C. Ennis, M. Hirsch and C. Pugh,*Foliations that are not approximable by smoother ones*, Report PAM-63, Center for Pure and Appl. Math., University of California, Berkeley, Calif., 1981.**[3]**Jenny Harrison,*Unsmoothable diffeomorphisms*, Ann. of Math. (2)**102**(1975), no. 1, 85–94. MR**0388458**, https://doi.org/10.2307/1970975**[4]**D. Hart,*On the smoothness of generators for flows and foliations*, Ph.D. Thesis, University of California, Berkeley, Calif., 1980.**[5]**Arthur J. Schwartz,*A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds*, Amer. J. Math. 85 (1963), 453-458; errata, ibid**85**(1963), 753. MR**0155061****[6]**Dennis Sullivan,*Hyperbolic geometry and homeomorphisms*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 543–555. MR**537749****[7]**F. Wesley Wilson Jr.,*Smoothing derivatives of functions and applications*, Trans. Amer. Math. Soc.**139**(1969), 413–428. MR**0251747**, https://doi.org/10.1090/S0002-9947-1969-0251747-9**[8]**F. Wesley Wilson Jr.,*Implicit submanifolds*, J. Math. Mech.**18**(1968/1969), 229–236. MR**0229252**

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0684511-4

Article copyright:
© Copyright 1983
American Mathematical Society